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 Electronic Communications in Probability > Vol. 6 (2001) > Paper 10 open journal systems 


On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

Jean Bertoin, Universite Pierre et Marie Curie
Marc Yor, Universite Pierre et Marie Curie


Abstract
Let $xi$ be a subordinator with Laplace exponent $Phi$, $I=int_{0}^{infty}exp(-xi_s)ds$ the so-called exponential functional, and $X$ (respectively, $hat X$) the self-similar Markov process obtained from $xi$ (respectively, from $hat{xi}=-xi$) by Lamperti's transformation. We establish the existence of a unique probability measure $rho$ on $]0,infty[$ with $k$-th moment given for every $kinN$ by the product $Phi(1)cdotsPhi(k)$, and which bears some remarkable connections with the preceding variables. In particular we show that if $R$ is an independent random variable with law $rho$ then $IR$ is a standard exponential variable, that the function $ttoE(1/X_t)$ coincides with the Laplace transform of $rho$, and that $rho$ is the $1$-invariant distribution of the sub-markovian process $hat X$. A number of known factorizations of an exponential variable are shown to be of the preceding form $IR$ for various subordinators $xi$.


Full text: PDF

Pages: 95-106

Published on: November 5, 2001
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Electronic Communications in Probability. ISSN: 1083-589X